Let $X$ and $Y$ be complex manifolds, let $f:Y\to X$ be a holomorphic map, and let $E\to X$ and $F\to X$ be holomorphic vector bundles. Let $p,q,r,s\in\mathbb{N}\cup\{0\}$. The Dolbeault operators
is induced from the tensor-product operator on $E^{\otimes k}$ and descends through the alternating quotient map $\operatorname{Alt}_k:E^{\otimes k}\to\Lambda^kE$, normalized by $\operatorname{Alt}_k(s_1\otimes\cdots\otimes s_k)=s_1\wedge\cdots\wedge s_k$. In particular, for $k\geq 1$ and smooth sections $s_1,\dots,s_k\in C^\infty(X,E)$,
For $k=0$, $\Lambda^0E$ is identified with the holomorphic line bundle $X\times\mathbb{C}$ and $\bar{\partial}_{\Lambda^0E}$ is the scalar Dolbeault operator. These induced operators are the unique ones compatible with evaluation, contraction, and the alternating quotient.